Download Practice Problems: Solving Diff. Equations & Applying Laplace Transform and more Exams Differential Equations in PDF only on Docsity! MA366 MIDTERM EXAM 2 – PRACTICE PROBLEMS 1. The general solution of y′′′ + 4y′′ + 5y′ = 0 is? 2. Find the solution of the initial value problem y(4) + 2y′′ + y = 3t + 4; y(0) = y′(0) = 0, y′′(0) = y′′′(0) = 1 by using the method of undetermined coefficients. 3. An object weighting 8 pounds attached to a spring will stretch it 6 inches beyond its natural length. There is a damping force with a damping constant c = 6 lbs- sec/ft and there is no external force. If at t = 0 the object is pulled 2 feet below equilibrium and then released, the initial value problem describing the vertical displacement x(t) becomes? 4. A spring-mass system is governed by the initial value problem x′′ + 4x′ + 4x = 4 cos ωt x(0) = 9, x′(0) = −2. For what value(s) of ω will resonance occur? 5. L{et(1 + cos 2t)} =? 6. Find the Laplace transform of f(t) = { t, 0 ≤ t < 1 0, 1 ≤ t < ∞ . 7. Solve y′′ + 3y′ + 2y = 4u1(t) y(0) = 0, y′(0) = 1. 8. Find the solution of the initial value problem y′′ + y = δ(t− π) y(0) = 0, y′(0) = 1. 9. The inverse Laplace transform of F (s) = se−s s2 + 2s + 5 is? 10. L {∫ t 0 sin 2(t− τ) cos(3τ)dτ } =? 1 2 MA366 MIDTERM EXAM 2 – PRACTICE PROBLEMS 11. Using the method of undetermined coefficients, determine the form of the particular solution yp(t) to the differential equation y′′′ − y′′ − y′ + y = 2t + 3et. A. yp(t) = 2t + at2et + bte−t B. yp(t) = at + b + ct2et C. yp(t) = at + bet + ctet + dt2 + et D. yp(t) = a + bt2et 12. If the characteristic equation has roots r = 1,−1+2i,−1−2i, with multiplicities 2, 1, 1 respectively, then a fundamental solution set is given by A. {et, e−t cos 2t, e−t sin 2t} B. {et, tet, e−t cos 2t, e−t sin 2t} C. {et, tet, e−2t cos t, e−2t sin t} D. {e−t, te−t, e−t cos 2t, e−t sin 2t} 13. Given f(t) = { 1 if 0 ≤ t < 1 t if 1 ≤ t , determine F (s) = L{f}. A. 1 s2 B. s + e−s s2 C. (s + 1)e−s s2 D. 1 s 14. Let y(t) be the solution of the initial value problem y′′ + 2y′ + 3y = δ(t− 1) + u2(t), y(0) = 0, y′(0) = 1. Compute the Laplace transform of y(t). A. Y (s) = 1 + e−s s2 + 2s + 3 + e−2s s(s2 + 2s + 3) B. Y (s) = e−s − 2 s2 + 2s + e−2s s(s2 + 2s) C. Y (s) = 2 + s + e−s s2 + 2s + 3 + e−2s s(s2 + 2s + 3) D. Y (s) = 1 + e−2s s2 + 2s + 3 + e−s s(s2 + 2s + 3) 15. Find the inverse Laplace transform of F (s) = 1− e−πs s(s2 + 1) . A. 1− cos t− uπ(t)(1 + cos t) B. (1− e−πt)(1− cos t) C. 1− cos(t)− δ(t− π) D. 1− cos t− uπ(t)(1− cos t)
